We now set up the proof of theorem:counterexample and
corollary:counterexample. We will construct a variation of Hodge
structure over the analytic moduli stack \({M}_{g,1}\)  whose restriction to each fiber of the forgetful map \({M}_{g,1}\rightarrow {M}_g\)  satisfies the hypotheses of lemma:positive-Hodge-bundle. We will do this via the Kodaira-Parshin trick
(see
[1]}
and also
[2]}),
which produces a family of curves over \({M}_{g,1}\)  which is non-isotrivial when restricted to each fiber of the natural forgetful map \({M}_{g,1}\rightarrow {M}_g\) .
We give a proof appealing to [3]},
but one can also prove it using prop:basic-facts and
corollary:unstable, as we mention in
remark:alternate-hodge-proof
Because we had difficulty finding a suitable reference, we now present a version
of the Kodaira-Parshin trick in families.
